For each x∈Xx\in X, there is only a finite number of j∈Jj\in J such that uj(x)≠0u_j(x) \neq 0 (point finiteness condition);

∑j∈Juj(x)=1\sum_{j \in J} u_j(x) = 1 for all x∈Xx\in X.

A partition of unity defines an open cover of XX, consisting of the open sets uj−1(0,1]u_j^{-1}(0,1]. Call this the induced cover.

Sometimes (rarely) the condition that {uj}J\{u_j\}_J is point finite is dropped. In this case we refer to a non-point finite partition of unity (see red herring principle). In this case for each point of XX at most countably-many of the functions uju_j are non-zero, and we have to interpret the sum in 1. above as being a convergent infinite series.

Given a cover𝒰={Uj}j∈J\mathcal{U} = \{U_j\}_{j\in J} of a topological space (open cover or closed or neither), the partition of unity {uj}J\{u_j\}_J is subordinate to 𝒰\mathcal{U} if for all j∈Jj\in J,

uj−1(0,1]¯⊂Uj.
\overline{u_j^{-1}(0,1]} \subset U_j.

What this means is that the open sets uj−1(0,1]u_j^{-1}(0,1] form an open cover refining the cover 𝒰\mathcal{U}.

Every open cover of (X,τ)(X,\tau) admits a subordinate partition of unity.

Similarly normal spaces are equivalently those such that every locally finite cover has a subordinate partition of unity (reference Bourbaki, Topology Generale - find this!)

Existence on smooth manifolds

Paracompact smooth manifolds even have smooth partitions of unity subordinate to any open cover (this follows from the existence of a smooth bump function on [−1,1][-1,1]). It is not true, however, that analytic manifolds have analytic partitions of unity - the aforementioned bump function is smooth but not analytic:

still form an open cover of XX. Hence for each point x∈Xx \in X there is i∈Ii \in I and j∈Jj \in J with x∈Ui∩Vjx \in U_i \cap V_j. By the nature of the Euclidean topology, there exists a closed ballBxB_x around ϕj−1(x)\phi_j^{-1}(x) in ϕj−1(Ui∩Vj)⊂ℝn\phi_j^{-1}(U_i \cap V_j) \subset \mathbb{R}^n. Its imageϕj(Bx)⊂X\phi_j(B_x) \subset X is a neighbourhood of x∈Xx \in X diffeomorphic to a closed ball.

is a finite cover by closed balls, hence in particular locally finite, and by construction it is still a refinement of the orignal cover. This shows the statement for XX compact.

Now for general XX, notice that without restriction we may assume that XX is connected, for if it is not, then we obtain the required refinement on all of XX by finding one on each connected component.

But if a locally Euclidean paracompact Hausdorff space XX is connected, then it is sigma-compact and in fact admits a countable increasing exhaustion

which canonically inherits the structure of a smooth manifold (this prop.). As above we find a refinement of the restriction of {Ui⊂X}i∈I\{U_i \subset X\}_{i \in I} to this open subset by closed balls and since the further subspace Kn+1∖KnK_{n+1}\setminus K_n is still compact (by this lemma) there is a finite set LnL_n such that

is a refinement by closed balls as required. Its local finiteness follows by the fact that each BlnB_{l_n} is contained in the “strip” Vn+2∖Kn−1V_{n+2} \setminus K_{n-1}, each strip contains only a finite set of BlnB_{l_n}-s and each strip intersects only a finite number of other strips. (Hence an open subset around a point xx which intersects only a finite number of elements of the refined cover is given by any one of the balls BlnB_{l_n} that contain xx.)

Proposition

Let XX be a paracompact smooth manifold. Then every open cover{Ui⊂X}i∈I\{U_i \subset X\}_{i \in I} has a subordinate partition of unity by functions {fi:Ui→ℝ}i∈I\{f_i \colon U_i \to \mathbb{R}\}_{i \in I} which are smooth functions.

is well defined (the sum involves only a finite number of non-vanishing contributions) and is smooth. Therefore setting

fj≔hjh
f_j \;\coloneqq\; \frac{h_j}{h}

then

{fj}j∈J
\left\{
f_j
\right\}_{j \in J}

is a subordinate partition of unity by smooth functions as required.

From a non-point finite partition of unity to a partition of unity

Definition

A collection of functions 𝒰={ui:X→[0,1]}\mathcal{U} = \{u_i : X \to [0,1]\} such that every x∈Xx\in X is in the support of some uiu_i. Then 𝒰\mathcal{U} is called locally finite if the cover ui−1(0,1]u_i^{-1}(0,1] (i.e. the induced cover) is locally finite.

Proposition

(Mather, 1965)

Let {ui}J\{u_i\}_J be a non-point finite partition of unity. Then there is a locally finite partition of unity {vi}i∈J\{v_i\}_{i\in J} such that the induced cover of the latter is a refinement of the induced cover of the former.

This implies that (loc. finite) numerable covers are cofinal in induced covers arising from collections of functions as in the definition. In particular, given the Milnor classifying spaceℬMG\mathcal{B}^M G of a topological groupGG, which comes with a countable family of ‘coordinate functions’ ℬMG→[0,1]\mathcal{B}^M G \to [0,1], has a numerable cover. This is shown by Dold to be a trivialising cover for the universal bundle constructed by Milnor, and so the universal bundle is numerable.

Coboundaries for Cech cocycles

Partitions of unity can be used to give explicit coboundaries for the cocycles of the complex of functions on a cover.

Let {Ui→X}\{U_i \to X\} be a open cover and {ρi∈C(X,ℝ)}\{\rho_i \in C(X,\mathbb{R})\} a collection of functions with

(xnot∈Ui)⇒ρi(x)=0(x not \in U_i) \Rightarrow \rho_i(x) = 0

∑iρi=const1\sum_i \rho_i = const_1.

Write C({Ui}):Δop→TopC(\{U_i\}) : \Delta^{op} \to Top for the Cech nerve of the cover and C(C({Ui}),ℝ)C(C(\{U_i\}), \mathbb{R}) for the cosimplicial ring of functions on this simplicial topological space; and (C•(C({Ui}),ℝ),δ)(C_\bullet(C(\{U_i\}), \mathbb{R}), \delta) for the corresponding (normalized) cochain complex: its differential is the alternating sum of the pullbacks of functions along the face maps, i.e. along the restriction maps